Sauté first 3 ingredients in hot oil 10 to 12 minutes or just until vegetables are tender and onion slices begin to turn golden.
Trim and discard rind from Brie.
Cut cheese into 1/2-inch cubes.
Layer a lightly greased 13- x 9-inch baking dish with half each of bread cubes, onion mixture, Brie cubes, and Parmesan cheese.
Whisk together eggs and next 4 ingredients; pour half of egg mixture evenly over cheeses.
Repeat layers once.
Cover and chill at least 8 hours or up to 24 hours.
Bake at 350° for 45 to 50 minutes or until lightly browned on top and set in center.
*2 cups (8 oz.) shredded Swiss cheese may be substituted.